Editing Ricci flow (section)

===Blow-up limits of singularities===

To study the formation of singularities it is useful, as in the study of other non-linear differential equations, to consider blow-ups limits. Intuitively speaking, one zooms into the singular region of the Ricci flow by rescaling time and space. Under certain assumptions, the zoomed in flow tends to a limiting Ricci flow <math> (M_\infty, g_\infty(t)), t \in (-\infty, 0] </math>, called a '''singularity model'''. Singularity models are ancient Ricci flows, i.e. they can be extended infinitely into the past. Understanding the possible singularity models in Ricci flow is an active research endeavor.

Below, we sketch the blow-up procedure in more detail: Let <math> (M, g_t), \, t \in [0,T), </math> be a Ricci flow that develops a singularity as <math>t \rightarrow T</math>. Let <math>(p_i, t_i) \in M \times [0,T) </math> be a sequence of points in spacetime such that 
:<math>K_i := \left|\operatorname{Rm}(g_{t_i})\right|(p_i) \rightarrow \infty  </math>
as <math>i \rightarrow \infty</math>. Then one considers the parabolically rescaled metrics
:<math>g_i(t) = K_i g\left(t_i + \frac{t}{K_i}\right), \quad t\in[-K_i t_i, 0]</math>
Due to the symmetry of the Ricci flow equation under parabolic dilations, the metrics <math>g_i(t)</math> are also solutions to the Ricci flow equation. In the case that
:<math> |Rm| \leq K_i  \text{ on } M \times [0,t_i],</math>
i.e. up to time <math>t_i</math> the maximum of the curvature is attained at <math>p_i</math>, then the pointed sequence of Ricci flows <math>(M, g_i(t), p_i)</math> subsequentially converges smoothly to a limiting ancient Ricci flow <math> (M_\infty, g_\infty(t), p_\infty)</math>. Note that in general <math> M_\infty </math> is not diffeomorphic to <math>M</math>.